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Question Number 21390 Answers: 0 Comments: 0

∫_0 ^1 ((x^7 − 1)/(ln x)) dx

$$\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}\:\:\frac{{x}^{\mathrm{7}} \:−\:\mathrm{1}}{\mathrm{ln}\:{x}}\:{dx} \\ $$

Question Number 21388 Answers: 0 Comments: 4

A block of mass m is connected with another block of mass 2m by a light spring. 2m is connected with a hanging mass 3m by an inextensible light string. At the time of release of block 3m, find tension in the string and acceleration of all the masses.

$$\mathrm{A}\:\mathrm{block}\:\mathrm{of}\:\mathrm{mass}\:{m}\:\mathrm{is}\:\mathrm{connected}\:\mathrm{with} \\ $$$$\mathrm{another}\:\mathrm{block}\:\mathrm{of}\:\mathrm{mass}\:\mathrm{2}{m}\:\mathrm{by}\:\mathrm{a}\:\mathrm{light} \\ $$$$\mathrm{spring}.\:\mathrm{2}{m}\:\mathrm{is}\:\mathrm{connected}\:\mathrm{with}\:\mathrm{a}\:\mathrm{hanging} \\ $$$$\mathrm{mass}\:\mathrm{3}{m}\:\mathrm{by}\:\mathrm{an}\:\mathrm{inextensible}\:\mathrm{light}\:\mathrm{string}. \\ $$$$\mathrm{At}\:\mathrm{the}\:\mathrm{time}\:\mathrm{of}\:\mathrm{release}\:\mathrm{of}\:\mathrm{block}\:\mathrm{3}{m},\:\mathrm{find} \\ $$$$\mathrm{tension}\:\mathrm{in}\:\mathrm{the}\:\mathrm{string}\:\mathrm{and}\:\mathrm{acceleration} \\ $$$$\mathrm{of}\:\mathrm{all}\:\mathrm{the}\:\mathrm{masses}. \\ $$

Question Number 21377 Answers: 0 Comments: 0

Balls are dropped from the roof of a tower at a fixed interval of time. At the moment when 9th ball reaches the ground the nth ball is (3/4)th height of the tower. What is the value of n?

$$\mathrm{Balls}\:\mathrm{are}\:\mathrm{dropped}\:\mathrm{from}\:\mathrm{the}\:\mathrm{roof}\:\mathrm{of}\:\mathrm{a} \\ $$$$\mathrm{tower}\:\mathrm{at}\:\mathrm{a}\:\mathrm{fixed}\:\mathrm{interval}\:\mathrm{of}\:\mathrm{time}.\:\mathrm{At}\:\mathrm{the} \\ $$$$\mathrm{moment}\:\mathrm{when}\:\mathrm{9th}\:\mathrm{ball}\:\mathrm{reaches}\:\mathrm{the} \\ $$$$\mathrm{ground}\:\mathrm{the}\:{n}\mathrm{th}\:\mathrm{ball}\:\mathrm{is}\:\left(\mathrm{3}/\mathrm{4}\right)\mathrm{th}\:\mathrm{height} \\ $$$$\mathrm{of}\:\mathrm{the}\:\mathrm{tower}.\:\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:{n}? \\ $$

Question Number 21366 Answers: 3 Comments: 0

Question Number 21357 Answers: 1 Comments: 0

Solve : log_(2x+3) x^2 < 1

$$\mathrm{Solve}\::\:\mathrm{log}_{\mathrm{2}{x}+\mathrm{3}} {x}^{\mathrm{2}} \:<\:\mathrm{1} \\ $$

Question Number 21356 Answers: 1 Comments: 0

Solve : (2^((3x−1)/(x−1)) )^(1/3) < 8^((x−3)/(3x−7))

$$\mathrm{Solve}\::\:\sqrt[{\mathrm{3}}]{\mathrm{2}^{\frac{\mathrm{3}{x}−\mathrm{1}}{{x}−\mathrm{1}}} }\:<\:\mathrm{8}^{\frac{{x}−\mathrm{3}}{\mathrm{3}{x}−\mathrm{7}}} \\ $$

Question Number 21355 Answers: 1 Comments: 0

Solve : ∣x^2 + 3x∣ + x^2 − 2 ≥ 0

$$\mathrm{Solve}\::\:\mid{x}^{\mathrm{2}} \:+\:\mathrm{3}{x}\mid\:+\:{x}^{\mathrm{2}} \:−\:\mathrm{2}\:\geqslant\:\mathrm{0} \\ $$

Question Number 21354 Answers: 0 Comments: 4

Solve : (√(2x + 5)) + (√(x − 1)) > 8

$$\mathrm{Solve}\::\:\sqrt{\mathrm{2}{x}\:+\:\mathrm{5}}\:+\:\sqrt{{x}\:−\:\mathrm{1}}\:>\:\mathrm{8} \\ $$

Question Number 21374 Answers: 0 Comments: 2

In how many ways can the letters of the word PATLIPUTRA be arranged, so that the relative order of vowels are consonants do not alter?

$$\mathrm{In}\:\mathrm{how}\:\mathrm{many}\:\mathrm{ways}\:\mathrm{can}\:\mathrm{the}\:\mathrm{letters}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{word}\:\mathrm{PATLIPUTRA}\:\mathrm{be}\:\mathrm{arranged},\:\mathrm{so} \\ $$$$\mathrm{that}\:\mathrm{the}\:\mathrm{relative}\:\mathrm{order}\:\mathrm{of}\:\mathrm{vowels}\:\mathrm{are} \\ $$$$\mathrm{consonants}\:\mathrm{do}\:\mathrm{not}\:\mathrm{alter}? \\ $$

Question Number 21350 Answers: 0 Comments: 0

prove (√2) < log_8 19 < (3)^(1/3)

$$\boldsymbol{{prove}}\: \\ $$$$\:\sqrt{\mathrm{2}}\:<\:\boldsymbol{{log}}_{\mathrm{8}} \mathrm{19}\:<\:\sqrt[{\mathrm{3}}]{\mathrm{3}} \\ $$

Question Number 21342 Answers: 1 Comments: 0

Question Number 21341 Answers: 2 Comments: 0

the angle between the straight lines x^2 +4xy+3y^2 =0 is

$${the}\:{angle}\:{between}\:{the}\:{straight}\:{lines}\:{x}^{\mathrm{2}} +\mathrm{4}{xy}+\mathrm{3}{y}^{\mathrm{2}} =\mathrm{0}\:{is} \\ $$

Question Number 21321 Answers: 1 Comments: 0

The number of real solutions of the equation 4x^(99) + 5x^(98) + 4x^(97) + 5x^(96) + ..... + 4x + 5 = 0 is

$$\mathrm{The}\:\mathrm{number}\:\mathrm{of}\:\mathrm{real}\:\mathrm{solutions}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{equation}\:\mathrm{4}{x}^{\mathrm{99}} \:+\:\mathrm{5}{x}^{\mathrm{98}} \:+\:\mathrm{4}{x}^{\mathrm{97}} \:+\:\mathrm{5}{x}^{\mathrm{96}} \:+ \\ $$$$.....\:+\:\mathrm{4}{x}\:+\:\mathrm{5}\:=\:\mathrm{0}\:\mathrm{is} \\ $$

Question Number 21319 Answers: 0 Comments: 0

If x, y, z are three real numbers such that x + y + z = 4 and x^2 + y^2 + z^2 = 6, then (1) (2/3) ≤ x, y, z ≤ 2 (2) 0 ≤ x, y, z ≤ 2 (3) 1 ≤ x, y, z ≤ 3 (4) 2 ≤ x, y, z ≤ 3

$$\mathrm{If}\:{x},\:{y},\:{z}\:\mathrm{are}\:\mathrm{three}\:\mathrm{real}\:\mathrm{numbers}\:\mathrm{such} \\ $$$$\mathrm{that}\:{x}\:+\:{y}\:+\:{z}\:=\:\mathrm{4}\:\mathrm{and}\:{x}^{\mathrm{2}} \:+\:{y}^{\mathrm{2}} \:+\:{z}^{\mathrm{2}} \:=\:\mathrm{6}, \\ $$$$\mathrm{then} \\ $$$$\left(\mathrm{1}\right)\:\frac{\mathrm{2}}{\mathrm{3}}\:\leqslant\:{x},\:{y},\:{z}\:\leqslant\:\mathrm{2} \\ $$$$\left(\mathrm{2}\right)\:\mathrm{0}\:\leqslant\:{x},\:{y},\:{z}\:\leqslant\:\mathrm{2} \\ $$$$\left(\mathrm{3}\right)\:\mathrm{1}\:\leqslant\:{x},\:{y},\:{z}\:\leqslant\:\mathrm{3} \\ $$$$\left(\mathrm{4}\right)\:\mathrm{2}\:\leqslant\:{x},\:{y},\:{z}\:\leqslant\:\mathrm{3} \\ $$

Question Number 21316 Answers: 0 Comments: 0

Let p = (x_1 − x_2 )^2 + (x_1 − x_3 )^2 + .... + (x_1 − x_6 )^2 + (x_2 − x_3 )^2 + (x_2 − x_4 )^2 + .... + (x_2 − x_6 )^2 + .... + (x_5 − x_6 )^2 = Σ_(1≤i<j≤6) ^6 (x_i − x_j )^2 . Then the maximum value of p if each x_i (i = 1, 2, ....., 6) has the value 0 and 1 is

Question Number 21315 Answers: 0 Comments: 0

The number of real solutions of the equation ((97 − x))^(1/4) + (x)^(1/4) = 5

$$\mathrm{The}\:\mathrm{number}\:\mathrm{of}\:\mathrm{real}\:\mathrm{solutions}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{equation}\:\sqrt[{\mathrm{4}}]{\mathrm{97}\:−\:{x}}\:+\:\sqrt[{\mathrm{4}}]{{x}}\:=\:\mathrm{5} \\ $$

Question Number 21314 Answers: 1 Comments: 0

Let α and β be the root of x^2 + px − (1/(2p^2 )) = 0, p ∈ R. The minimum value of α^4 + β^4 is

$$\mathrm{Let}\:\alpha\:\mathrm{and}\:\beta\:\mathrm{be}\:\mathrm{the}\:\mathrm{root}\:\mathrm{of}\:{x}^{\mathrm{2}} \:+\:{px}\:−\:\frac{\mathrm{1}}{\mathrm{2}{p}^{\mathrm{2}} }\:=\:\mathrm{0}, \\ $$$${p}\:\in\:{R}.\:\mathrm{The}\:\mathrm{minimum}\:\mathrm{value}\:\mathrm{of}\:\alpha^{\mathrm{4}} \:+\:\beta^{\mathrm{4}} \:\mathrm{is} \\ $$

Question Number 21313 Answers: 0 Comments: 4

Let k be a real number such that the inequality (√(x − 3)) + (√(6 − x)) ≥ k has a solution then the maximum value of k is

$$\mathrm{Let}\:{k}\:\mathrm{be}\:\mathrm{a}\:\mathrm{real}\:\mathrm{number}\:\mathrm{such}\:\mathrm{that}\:\mathrm{the} \\ $$$$\mathrm{inequality}\:\sqrt{{x}\:−\:\mathrm{3}}\:+\:\sqrt{\mathrm{6}\:−\:{x}}\:\geqslant\:{k}\:\mathrm{has}\:\mathrm{a} \\ $$$$\mathrm{solution}\:\mathrm{then}\:\mathrm{the}\:\mathrm{maximum}\:\mathrm{value}\:\mathrm{of}\:{k} \\ $$$$\mathrm{is} \\ $$

Question Number 21311 Answers: 0 Comments: 0

Let a and b be positive real numbers with a^3 + b^3 = a − b, and k = a^2 + 4b^2 , then (1) k < 1 (2) k >1 (3) k = 1 (4) k > 2

$$\mathrm{Let}\:{a}\:\mathrm{and}\:{b}\:\mathrm{be}\:\mathrm{positive}\:\mathrm{real}\:\mathrm{numbers} \\ $$$$\mathrm{with}\:{a}^{\mathrm{3}} \:+\:{b}^{\mathrm{3}} \:=\:{a}\:−\:{b},\:\mathrm{and}\:{k}\:=\:{a}^{\mathrm{2}} \:+\:\mathrm{4}{b}^{\mathrm{2}} , \\ $$$$\mathrm{then} \\ $$$$\left(\mathrm{1}\right)\:{k}\:<\:\mathrm{1} \\ $$$$\left(\mathrm{2}\right)\:{k}\:>\mathrm{1} \\ $$$$\left(\mathrm{3}\right)\:{k}\:=\:\mathrm{1} \\ $$$$\left(\mathrm{4}\right)\:{k}\:>\:\mathrm{2} \\ $$

Question Number 21310 Answers: 1 Comments: 8

What do you guys think of creating a Telegram group to discuss theory and more descriptive questions?

$$\mathrm{What}\:\mathrm{do}\:\mathrm{you}\:\mathrm{guys}\:\mathrm{think}\:\mathrm{of} \\ $$$$\mathrm{creating}\:\mathrm{a}\:\mathrm{Telegram}\:\mathrm{group}\:\mathrm{to} \\ $$$$\mathrm{discuss}\:\mathrm{theory}\:\mathrm{and}\:\mathrm{more}\:\mathrm{descriptive} \\ $$$$\mathrm{questions}? \\ $$

Question Number 21309 Answers: 0 Comments: 0

Suppose p is a polynomial with complex coefficients and an even degree. If all the roots of p are complex non-real numbers with modulus 1, prove that p(1) ∈ R iff p(−1) ∈ R.

$$\mathrm{Suppose}\:{p}\:\mathrm{is}\:\mathrm{a}\:\mathrm{polynomial}\:\mathrm{with}\:\mathrm{complex} \\ $$$$\mathrm{coefficients}\:\mathrm{and}\:\mathrm{an}\:\mathrm{even}\:\mathrm{degree}.\:\mathrm{If}\:\mathrm{all} \\ $$$$\mathrm{the}\:\mathrm{roots}\:\mathrm{of}\:{p}\:\mathrm{are}\:\mathrm{complex}\:\mathrm{non}-\mathrm{real} \\ $$$$\mathrm{numbers}\:\mathrm{with}\:\mathrm{modulus}\:\mathrm{1},\:\mathrm{prove}\:\mathrm{that} \\ $$$${p}\left(\mathrm{1}\right)\:\in\:{R}\:\mathrm{iff}\:{p}\left(−\mathrm{1}\right)\:\in\:{R}. \\ $$

Question Number 21308 Answers: 0 Comments: 0

Find all complex numbers z such that ∣z − ∣z + 1∣∣ = ∣z + ∣z − 1∣∣

$$\mathrm{Find}\:\mathrm{all}\:\mathrm{complex}\:\mathrm{numbers}\:{z}\:\mathrm{such}\:\mathrm{that} \\ $$$$\mid{z}\:−\:\mid{z}\:+\:\mathrm{1}\mid\mid\:=\:\mid{z}\:+\:\mid{z}\:−\:\mathrm{1}\mid\mid \\ $$

Question Number 21307 Answers: 0 Comments: 5

Let z_1 , z_2 , z_3 be complex numbers such that (i) ∣z_1 ∣ = ∣z_2 ∣ = ∣z_3 ∣ = 1 (ii) z_1 + z_2 + z_3 ≠ 0 (iii) z_1 ^2 + z_2 ^2 + z_3 ^2 = 0 Prove that for all n ≥ 2, ∣z_1 ^n + z_2 ^n + z_3 ^n ∣ ∈ {0, 1, 2, 3}.

$$\mathrm{Let}\:{z}_{\mathrm{1}} ,\:{z}_{\mathrm{2}} ,\:{z}_{\mathrm{3}} \:\mathrm{be}\:\mathrm{complex}\:\mathrm{numbers}\:\mathrm{such} \\ $$$$\mathrm{that} \\ $$$$\left(\mathrm{i}\right)\:\mid{z}_{\mathrm{1}} \mid\:=\:\mid{z}_{\mathrm{2}} \mid\:=\:\mid{z}_{\mathrm{3}} \mid\:=\:\mathrm{1} \\ $$$$\left(\mathrm{ii}\right)\:{z}_{\mathrm{1}} \:+\:{z}_{\mathrm{2}} \:+\:{z}_{\mathrm{3}} \:\neq\:\mathrm{0} \\ $$$$\left(\mathrm{iii}\right)\:{z}_{\mathrm{1}} ^{\mathrm{2}} \:+\:{z}_{\mathrm{2}} ^{\mathrm{2}} \:+\:{z}_{\mathrm{3}} ^{\mathrm{2}} \:=\:\mathrm{0} \\ $$$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{for}\:\mathrm{all}\:{n}\:\geqslant\:\mathrm{2}, \\ $$$$\mid{z}_{\mathrm{1}} ^{{n}} \:+\:{z}_{\mathrm{2}} ^{{n}} \:+\:{z}_{\mathrm{3}} ^{{n}} \mid\:\in\:\left\{\mathrm{0},\:\mathrm{1},\:\mathrm{2},\:\mathrm{3}\right\}. \\ $$

Question Number 21295 Answers: 1 Comments: 0

For a particle performing uniform circular motion, angular momentum is constant in magnitude but direction keeps changing. Am I right or wrong?

$$\mathrm{For}\:\mathrm{a}\:\mathrm{particle}\:\mathrm{performing}\:\mathrm{uniform} \\ $$$$\mathrm{circular}\:\mathrm{motion},\:\mathrm{angular}\:\mathrm{momentum}\:\mathrm{is} \\ $$$$\mathrm{constant}\:\mathrm{in}\:\mathrm{magnitude}\:\mathrm{but}\:\mathrm{direction} \\ $$$$\mathrm{keeps}\:\mathrm{changing}. \\ $$$$\mathrm{Am}\:\mathrm{I}\:\mathrm{right}\:\mathrm{or}\:\mathrm{wrong}? \\ $$

Question Number 21294 Answers: 1 Comments: 0

Let z_1 , z_2 , z_3 be complex numbers, not all real, such that ∣z_1 ∣ = ∣z_2 ∣ = ∣z_3 ∣ = 1 and 2(z_1 + z_2 + z_3 ) − 3z_1 z_2 z_3 ∈ R. Prove that max(arg z_1 , arg z_2 , arg z_3 ) ≥ (π/6) . Where 0 < arg(z_1 ), arg(z_2 ), arg(z_3 ) < 2π.

$$\mathrm{Let}\:{z}_{\mathrm{1}} ,\:{z}_{\mathrm{2}} ,\:{z}_{\mathrm{3}} \:\mathrm{be}\:\mathrm{complex}\:\mathrm{numbers},\:\mathrm{not} \\ $$$$\mathrm{all}\:\mathrm{real},\:\mathrm{such}\:\mathrm{that}\:\mid{z}_{\mathrm{1}} \mid\:=\:\mid{z}_{\mathrm{2}} \mid\:=\:\mid{z}_{\mathrm{3}} \mid\:=\:\mathrm{1} \\ $$$$\mathrm{and}\:\mathrm{2}\left({z}_{\mathrm{1}} \:+\:{z}_{\mathrm{2}} \:+\:{z}_{\mathrm{3}} \right)\:−\:\mathrm{3}{z}_{\mathrm{1}} {z}_{\mathrm{2}} {z}_{\mathrm{3}} \:\in\:{R}. \\ $$$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{max}\left(\mathrm{arg}\:{z}_{\mathrm{1}} ,\:\mathrm{arg}\:{z}_{\mathrm{2}} ,\:\mathrm{arg}\:{z}_{\mathrm{3}} \right)\:\geqslant \\ $$$$\frac{\pi}{\mathrm{6}}\:.\:\mathrm{Where}\:\mathrm{0}\:<\:\mathrm{arg}\left({z}_{\mathrm{1}} \right),\:\mathrm{arg}\left({z}_{\mathrm{2}} \right),\:\mathrm{arg}\left({z}_{\mathrm{3}} \right) \\ $$$$<\:\mathrm{2}\pi. \\ $$

Question Number 21293 Answers: 1 Comments: 0

Let n be an even positive integer such that (n/2) is odd and let α_0 , α_1 , ...., α_(n−1) be the complex roots of unity of order n. Prove that Π_(k=0) ^(n−1) (a + bα_k ^2 ) = (a^(n/2) + b^(n/2) )^2 for any complex numbers a and b.

$$\mathrm{Let}\:{n}\:\mathrm{be}\:\mathrm{an}\:\mathrm{even}\:\mathrm{positive}\:\mathrm{integer}\:\mathrm{such} \\ $$$$\mathrm{that}\:\frac{{n}}{\mathrm{2}}\:\mathrm{is}\:\mathrm{odd}\:\mathrm{and}\:\mathrm{let}\:\alpha_{\mathrm{0}} ,\:\alpha_{\mathrm{1}} ,\:....,\:\alpha_{{n}−\mathrm{1}} \:\mathrm{be} \\ $$$$\mathrm{the}\:\mathrm{complex}\:\mathrm{roots}\:\mathrm{of}\:\mathrm{unity}\:\mathrm{of}\:\mathrm{order}\:{n}. \\ $$$$\mathrm{Prove}\:\mathrm{that}\:\underset{{k}=\mathrm{0}} {\overset{{n}−\mathrm{1}} {\prod}}\left({a}\:+\:{b}\alpha_{{k}} ^{\mathrm{2}} \right)\:=\:\left({a}^{\frac{{n}}{\mathrm{2}}} \:+\:{b}^{\frac{{n}}{\mathrm{2}}} \right)^{\mathrm{2}} \\ $$$$\mathrm{for}\:\mathrm{any}\:\mathrm{complex}\:\mathrm{numbers}\:{a}\:\mathrm{and}\:{b}. \\ $$

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